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Numerical Solution of the Navier-Stokes Equations for Incompressible Fluid Flow by Crank-Nicolson Implicit Scheme

Received: 18 January 2021     Accepted: 26 January 2021     Published: 12 March 2021
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Abstract

The Navier-Stokes (N-S) equations for incompressible fluid flow comprise of a system of four nonlinear equations with five flow fields such as pressure P, density ρ and three velocity components u, v, and w. The system of equations is generally complex due to the fact that it is nonlinear and a mixture of the three classes of partial differential equations (PDEs) each with distinct solution methods. The N-S equations fully describe the unsteady fluid flow behaviour of laminar and turbulent types. Previous studies have shown existence of general solutions of fluid flow models but little has been done on numerical solution for velocity of flow in N-S equation of incompressible fluid flow by Crank-Nicolson implicit scheme. In practice, real fluid flows are compressible due to the inevitable variations in density caused by temperature changes and other physical factors. Numerical approximations of the general system of Navier-Stokes equations were made to develop numerical solution model for incompressible fluid flow. Adequate solutions of the latter produce numerical solutions applicable in numerical simulation of fluid flows useful in engineering and science. Non-dimensionalization of variables involved was done. Crank-Nicolson (C.N) implicit scheme was implemented to discretize partial derivatives and appropriate approximation made at the boundaries yielded a linear system of N-S equations model. The linear numerical system was then expressed in matrix form for computation of velocity field by Computational fluid dynamics (CFD) approach using MATLAB software. Numerical results for velocity field in two dimensional space, u(x,y,t)and v(x,y,t) generated in uniform 32×32 grids points of the square flow domains, 0≤x≤1.0 and 0≤y≤1.0 were presented in three dimensional figures. Results showed that the velocity in two dimensional space does not change suddenly for any change in spatial levels, x and y. Therefore, C-N implicit Scheme applied to solve the N-S equations for fluid flow is consistent.

Published in Applied and Computational Mathematics (Volume 10, Issue 1)
DOI 10.11648/j.acm.20211001.12
Page(s) 10-18
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This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2021. Published by Science Publishing Group

Keywords

Navier- Stokes Equation, Nonlinear System, Incompressible Fluid, Vorticity, Coriolis Force, Discretization

References
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[2] Christov C. I. & Marinova R.S (2001). Implicit Vectorial Operator Splitting for Incompressible Navier- Stokes Equations in Primitive Variables. Vol 6, No. 4
[3] Cohen N. & Pijush K Kundu. 4th Edition. (2008). Fluid mechanics. (Elsevier)
[4] Fadugha S. & Zelibe S.C. (2015). Crank-Nicolson method for solving parabolic partial differential equations. Adoekiti University (Nigeria). c/oi http//www.researchgate.net/pub/ 287195203.http://doi.org./10.1016/jcp.2006.01.036
[5] Fernandes Enrique-Cara & Irene Marin I-Gayte. (2018) A New Proof of Existence of Suitable Weak Solution and Other Remarks for Navier-Stokes Equations. Applied Mathematics, 9,383-402. http://doi.org./10.4236/am.2018.94029
[6] Jiyuan Tu, Guan Heng Y & Chaoqun Lui. 2nd Edition. (2013). Computational fluid dynamics: a practical approach. (b) (Elsevier)
[7] Kanti Pandey. & Verma Lijja.(2011). A note on Crank-Nicolson for Burger’s Equation. Applied Mathematics. 2011, 2,883-889. http://doi.org./10.4236./am.2011.27118
[8] Kweyu C.M. Nyamai B. M., Wahome J. N., Kandie J. N. & Bundotich J. K. (2013). Modified Crank-Nicolson scheme for Numerical solutions of the 2D coupled Burger’s system. Moi university (Kenya) c/oi http//www.researchgate.net/pub/ 263100789
[9] Loukopoulos V. C, Messaris G. T & Bourantas G. C. (2011). Numerical solution for incompressible Navier -Stokes Equations in primitive Variables and velocity- vorticity formulations: Journal of Applied Mathematics and computations, 222 (2013) 575-588.
[10] Munson B.R, Donald F.Y, Theodore H.O. & Wade W.H (2009). Fundamentals of Fluid Mechanics. (Wiley)
[11] Nakolay Nikitin V. (2006). Finite difference Methods for Incompressible Navier-Stokes Equations in Arbitrary Orthogonal Curvilinear coordinates. Journal of Computational Physics. 217(2006) 759-781.
[12] Panton L. Ronald 4th Edition (2013) Incompressible flow. (Wiley).
[13] Popescu Ioana. (2014). Computational hydraulics. Numerical methods and modelling. (IWA Publishing)
[14] Maritim S., Rotich J. K, & Bitok J.K. (2019). Hybrid Hopscotch Method for Solving Two Dimensional System of Burgers’ equations. International Journal of Science and Research. http://doi.org./10.21275/ART.2020.186 ISSN: 2319-7064
[15] Saqib M., Shahid H. & Daoud S. M (2017).Computational solutions of Two-dimensional Convection-Diffussion Equations Using Crank-Nicolson Implicit and Time Efficient ADI. American Journal of Computational Mathematics. 2017, 7, 208-227. http://doi.org./10.4236/ajcm.2017.73019
[16] Schezt J.A & Rodney D.W.P. (2011) boundary layer analysis. (AIAA Education series)
[17] Stampolidis P & Guosiduo-Kouttita M.C. (2008). A computational Study with Finite Difference Methods for second order Quasilinear Hyperbolic Partial Differential Equations in Two Independent Variables. Applied mathematics, 2018, 9, 1193-1224.
[18] Vineet Kumar S., Muhamed T., Utkarsh B. & Sanyasiraju YUSS (2011). Crank-Nicolson Scheme for Numerical Solution of Two Dimensional Coupled Burger’s Equations. International journal for scientific Engineering and Research. http://www.ijser.org
[19] Zikanov Oleg (2010). Essentials of computational Fluid Dynamics. (Wiley)
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    Siele Charles, Rotich John, Adicka Daniel. (2021). Numerical Solution of the Navier-Stokes Equations for Incompressible Fluid Flow by Crank-Nicolson Implicit Scheme. Applied and Computational Mathematics, 10(1), 10-18. https://doi.org/10.11648/j.acm.20211001.12

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    Siele Charles; Rotich John; Adicka Daniel. Numerical Solution of the Navier-Stokes Equations for Incompressible Fluid Flow by Crank-Nicolson Implicit Scheme. Appl. Comput. Math. 2021, 10(1), 10-18. doi: 10.11648/j.acm.20211001.12

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    AMA Style

    Siele Charles, Rotich John, Adicka Daniel. Numerical Solution of the Navier-Stokes Equations for Incompressible Fluid Flow by Crank-Nicolson Implicit Scheme. Appl Comput Math. 2021;10(1):10-18. doi: 10.11648/j.acm.20211001.12

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  • @article{10.11648/j.acm.20211001.12,
      author = {Siele Charles and Rotich John and Adicka Daniel},
      title = {Numerical Solution of the Navier-Stokes Equations for Incompressible Fluid Flow by Crank-Nicolson Implicit Scheme},
      journal = {Applied and Computational Mathematics},
      volume = {10},
      number = {1},
      pages = {10-18},
      doi = {10.11648/j.acm.20211001.12},
      url = {https://doi.org/10.11648/j.acm.20211001.12},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20211001.12},
      abstract = {The Navier-Stokes (N-S) equations for incompressible fluid flow comprise of a system of four nonlinear equations with five flow fields such as pressure P, density ρ and three velocity components u, v, and w. The system of equations is generally complex due to the fact that it is nonlinear and a mixture of the three classes of partial differential equations (PDEs) each with distinct solution methods. The N-S equations fully describe the unsteady fluid flow behaviour of laminar and turbulent types. Previous studies have shown existence of general solutions of fluid flow models but little has been done on numerical solution for velocity of flow in N-S equation of incompressible fluid flow by Crank-Nicolson implicit scheme. In practice, real fluid flows are compressible due to the inevitable variations in density caused by temperature changes and other physical factors. Numerical approximations of the general system of Navier-Stokes equations were made to develop numerical solution model for incompressible fluid flow. Adequate solutions of the latter produce numerical solutions applicable in numerical simulation of fluid flows useful in engineering and science. Non-dimensionalization of variables involved was done. Crank-Nicolson (C.N) implicit scheme was implemented to discretize partial derivatives and appropriate approximation made at the boundaries yielded a linear system of N-S equations model. The linear numerical system was then expressed in matrix form for computation of velocity field by Computational fluid dynamics (CFD) approach using MATLAB software. Numerical results for velocity field in two dimensional space, u(x,y,t)and v(x,y,t) generated in uniform 32×32 grids points of the square flow domains, 0≤x≤1.0 and 0≤y≤1.0 were presented in three dimensional figures. Results showed that the velocity in two dimensional space does not change suddenly for any change in spatial levels, x and y. Therefore, C-N implicit Scheme applied to solve the N-S equations for fluid flow is consistent.},
     year = {2021}
    }
    

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  • TY  - JOUR
    T1  - Numerical Solution of the Navier-Stokes Equations for Incompressible Fluid Flow by Crank-Nicolson Implicit Scheme
    AU  - Siele Charles
    AU  - Rotich John
    AU  - Adicka Daniel
    Y1  - 2021/03/12
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    N1  - https://doi.org/10.11648/j.acm.20211001.12
    DO  - 10.11648/j.acm.20211001.12
    T2  - Applied and Computational Mathematics
    JF  - Applied and Computational Mathematics
    JO  - Applied and Computational Mathematics
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    EP  - 18
    PB  - Science Publishing Group
    SN  - 2328-5613
    UR  - https://doi.org/10.11648/j.acm.20211001.12
    AB  - The Navier-Stokes (N-S) equations for incompressible fluid flow comprise of a system of four nonlinear equations with five flow fields such as pressure P, density ρ and three velocity components u, v, and w. The system of equations is generally complex due to the fact that it is nonlinear and a mixture of the three classes of partial differential equations (PDEs) each with distinct solution methods. The N-S equations fully describe the unsteady fluid flow behaviour of laminar and turbulent types. Previous studies have shown existence of general solutions of fluid flow models but little has been done on numerical solution for velocity of flow in N-S equation of incompressible fluid flow by Crank-Nicolson implicit scheme. In practice, real fluid flows are compressible due to the inevitable variations in density caused by temperature changes and other physical factors. Numerical approximations of the general system of Navier-Stokes equations were made to develop numerical solution model for incompressible fluid flow. Adequate solutions of the latter produce numerical solutions applicable in numerical simulation of fluid flows useful in engineering and science. Non-dimensionalization of variables involved was done. Crank-Nicolson (C.N) implicit scheme was implemented to discretize partial derivatives and appropriate approximation made at the boundaries yielded a linear system of N-S equations model. The linear numerical system was then expressed in matrix form for computation of velocity field by Computational fluid dynamics (CFD) approach using MATLAB software. Numerical results for velocity field in two dimensional space, u(x,y,t)and v(x,y,t) generated in uniform 32×32 grids points of the square flow domains, 0≤x≤1.0 and 0≤y≤1.0 were presented in three dimensional figures. Results showed that the velocity in two dimensional space does not change suddenly for any change in spatial levels, x and y. Therefore, C-N implicit Scheme applied to solve the N-S equations for fluid flow is consistent.
    VL  - 10
    IS  - 1
    ER  - 

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Author Information
  • Department of Mathematics and Computer Science, School of Science and Technology, University of Kabianga, Kericho, Kenya

  • Department of Mathematics and Computer Science, School of Science and Technology, University of Kabianga, Kericho, Kenya

  • Department of Mathematics and Computer Science, School of Science and Technology, University of Kabianga, Kericho, Kenya

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